Place the rice and 3 cups water in a medium saucepan set over high heat. Cover, and bring to a boil; reduce heat to low, and cook until rice is tender, about 15 minutes. Turn off the heat, uncover, and sprinkle rice with 1/4 cup mirin. Replace lid, and allow rice to sit for 15 minutes. Transfer rice to a medium bowl, and let cool to room temperature.
To make the rolls: Lay a sheet of nori on a bamboo sushi roller, and sprinkle with mirin. Spread 1 1/4 cups cooked rice on the nori, leaving a 1 1/2-inch border on each long side. Spread 1/4 teaspoon wasabi paste in a horizontal strip across the middle of the rice. Place 4 to 5 carrot, red-pepper, celery, and scallion matchsticks on top of the wasabi paste. Cover with a layer of ginger. Lift the closest end of the bamboo roller up and over the vegetables so the 2 long edges of nori meet. Press the roll firmly inside the bamboo roller. Transfer roll to a plate, cover with a damp towel, and place in the refrigerator. Continue to roll sushi until all ingredients have been used.
Remove rolls from refrigerator; using a sharp knife, trim ends and cut rolls into 8 equal pieces. Serve with pickled ginger and dipping sauce.